3.108 \(\int \frac {1}{\sqrt {a+b \sinh (x)}} \, dx\)

Optimal. Leaf size=60 \[ \frac {2 i \sqrt {\frac {a+b \sinh (x)}{a-i b}} F\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{\sqrt {a+b \sinh (x)}} \]

[Out]

2*I*(sin(1/4*Pi+1/2*I*x)^2)^(1/2)/sin(1/4*Pi+1/2*I*x)*EllipticF(cos(1/4*Pi+1/2*I*x),2^(1/2)*(b/(I*a+b))^(1/2))
*((a+b*sinh(x))/(a-I*b))^(1/2)/(a+b*sinh(x))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2663, 2661} \[ \frac {2 i \sqrt {\frac {a+b \sinh (x)}{a-i b}} F\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{\sqrt {a+b \sinh (x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + b*Sinh[x]],x]

[Out]

((2*I)*EllipticF[Pi/4 - (I/2)*x, (2*b)/(I*a + b)]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/Sqrt[a + b*Sinh[x]]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2663

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a+b \sinh (x)}} \, dx &=\frac {\sqrt {\frac {a+b \sinh (x)}{a-i b}} \int \frac {1}{\sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}}} \, dx}{\sqrt {a+b \sinh (x)}}\\ &=\frac {2 i F\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{\sqrt {a+b \sinh (x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.19, size = 60, normalized size = 1.00 \[ \frac {2 i \sqrt {\frac {a+b \sinh (x)}{a-i b}} F\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right )}{\sqrt {a+b \sinh (x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a + b*Sinh[x]],x]

[Out]

((2*I)*EllipticF[(Pi - (2*I)*x)/4, ((-2*I)*b)/(a - I*b)]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/Sqrt[a + b*Sinh[x]]

________________________________________________________________________________________

fricas [F]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {b \sinh \relax (x) + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sinh(x))^(1/2),x, algorithm="fricas")

[Out]

integral(1/sqrt(b*sinh(x) + a), x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \sinh \relax (x) + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sinh(x))^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(b*sinh(x) + a), x)

________________________________________________________________________________________

maple [A]  time = 0.10, size = 125, normalized size = 2.08 \[ -\frac {2 \left (i b -a \right ) \sqrt {-\frac {a +b \sinh \relax (x )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \relax (x )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \relax (x )\right ) b}{i b -a}}\, \EllipticF \left (\sqrt {-\frac {a +b \sinh \relax (x )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right )}{b \cosh \relax (x ) \sqrt {a +b \sinh \relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sinh(x))^(1/2),x)

[Out]

-2*(I*b-a)*(-(a+b*sinh(x))/(I*b-a))^(1/2)*((I-sinh(x))*b/(I*b+a))^(1/2)*((I+sinh(x))*b/(I*b-a))^(1/2)*Elliptic
F((-(a+b*sinh(x))/(I*b-a))^(1/2),(-(I*b-a)/(I*b+a))^(1/2))/b/cosh(x)/(a+b*sinh(x))^(1/2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \sinh \relax (x) + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sinh(x))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(b*sinh(x) + a), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {a+b\,\mathrm {sinh}\relax (x)}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*sinh(x))^(1/2),x)

[Out]

int(1/(a + b*sinh(x))^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + b \sinh {\relax (x )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sinh(x))**(1/2),x)

[Out]

Integral(1/sqrt(a + b*sinh(x)), x)

________________________________________________________________________________________